“I’ve got” seems particularly strange to me because without the contraction Americans would still just say “I have.” (There are some circumstances where they’ll say “I have got” without a contraction, but it’s mainly when they’re drawing a contrast with what they “haven’t got.” E.g., “No, I don’t have a baseball… oh, but I have got a lacrosse ball, will that work?”)

I think the rule is probably closer to “you don’t contract a stressed verb,” but that’s not terribly useful since there are so few rules about stress patterns. Verbs at the end of sentences are typically stressed, though, so you’re right that ending with that kind of contraction is going to sound wrong to most people.